Algebra and geometry of link homology

Abstract

These notes cover the lectures of the first named author at 2021 IHES Summer School on "Enumerative Geometry, Physics and Representation Theory" with additional details and references. They cover the definition of Khovanov-Rozansky triply graded homology, its basic properties and recent advances, as well as three algebro-geometric models for link homology: braid varieties, Hilbert schemes of singular curves and affine Springer fibers, and Hilbert schemes of points on the plane.

arXiv comment49 pages

References in corpus (24)

• Vanishing theorems and character formulas for the Hilbert scheme of points in the plane
• Quantizations of conical symplectic resolutions I: local and global structure
• Legendrian knots and constructible sheaves
• The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link
• Toric braids and $(m,n)$-parking functions
• Categorification of the braid groups
• Homology of torus knots
• Quantized Coulomb branches of Jordan quiver gauge theories and cyclotomic rational Cherednik algebras
• Cohomology of cluster varieties. I. Locally acyclic case
• A geometric model for Hochschild homology of Soergel bimodules
• Hilbert schemes and $y$-ification of Khovanov-Rozansky homology
• Categorified Young symmetrizers and stable homology of torus links
• Khovanov-Rozansky homology and higher Catalan sequences
• Cell decompositions of character varieties
• Augmentations, Fillings, and Clusters
• BFN Springer Theory
• Torus link homology
• Dualizable link homology
• From the Hecke Category to the Unipotent Locus
• The cohomology ring of certain compactified Jacobians
• Soergel bimodules and matrix factorizations
• Tautological classes and symmetry in Khovanov-Rozansky homology
• A categorification of a cyclotomic Hecke algebra
• Affine Springer Fibers, Procesi bundles, and Cherednik algebras